R.A. Broglia
Milano University and INFN
The Niels Bohr Institute, Copenhagen
F. Barranco and G. Potel
Sevilla University
Medium polarization effects and transfer reactions in halo nuclei
E. Vigezzi
INFN Milano
-The dynamic halo
- Microscopic description of two nucleon transfer reactions
Outline
Mean-field results(Sagawa,Brown,Esbensen PLB 309(93)1)Experimental systematics
Parity inversion in N=7 isotones
Eshift = - 2.5 MeV
p1/2p3/2
2+
2+
s1/2
Eshift = + 2.5 MeV
s1/2
d5/2
5 MeV 0 MeV
½ -
½+
Pauli blocking of core ground state correlations
Self-energy
Level inversion
+
11Be
ja
jb
λ
SELF ENERGY RENORMALIZATION OF SINGLE-PARTICLE STATES: CLOSED SHELL
C. Mahaux, P.F. Bortignon, R.A. Broglia, C.H. Dasso and Mahaux, Phys. Rep.(1985)1
208Pb
e1
e2 =
=
e2
e1
e2
e1e1
Calculated ground state
213.087.02/1 2/52/1 ds
Exp.:
J.S. Winfield et al., Nucl.Phys. A683 (2001) 48
216.084.02/1 2/52/1 ds
11Be(p,d)10Be in inverse kinematic detecting both the ground state and the 2+ excited state of 10Be.
Fermionic degrees of freedom:
• s1/2, p1/2, d5/2 Wood-Saxon levels up to 150 MeV (discretized continuum) from a standard (Bohr-Mottelson) Woods-Saxon potential
Bosonic degrees of freedom:
• 2+ and 3- QRPA solutions with energy up to 50 MeV; residual
interaction: multipole-multipole separable with the coupling
constant tuned to reproduce E(2+)=3.36 MeV and 0.6<2<0.7
Main ingredients of our calculation
11Be
New result for S[1/2+]: 0.28+0.03 -0.07
Kanungo et al.PLB 682 (2010) 39
Spectroscopic factors from (12Be,11Be+γ)reaction to ½+ and ½- final states:S[1/2-]= 0.37±0.10 S[1/2+]= 0.42±0.10
Good agreement between theory and experiment concerning energies and “spectroscopic” factors
Theoretical calculation for 11Li
Low-lying dipole strength
s-p mixing
0.369 MeV
A dynamical description of two-neutron halos
11LiF. Barranco et al. EPJ A11 (2001) 385
12BeG. Gori et al. PRC 69 (2004) 041302(R)
Induced interaction
Energy-dependent matrix
Bare interaction
Correlated halo wavefunction
Uncorrelated
H. Esbensen, B.A. Brown, H. Sagawa, PRC 51 (1995) 1274
Vibrational vs. deformed core
Two-neutron transfer to
ground state
exc. state
How to probe the particle-phonon coupling?Test the microscopic correlated wavefunction with phonon admixture
r
Probing 11Li halo-neutrons correlationsvia (p,t) reaction
Calculation of absolute two-nucleon transfer cross section by finite-range DWBA calculation
B.F. Bayman and J. Chen, Phys. Rev. C 26 (1982) 150M. Igarashi, K. Kubo and K. Yagi, Phys. Rep. 199 (1991) 1G. Potel et al., arXiv:0906.4298
G. Potel et al., arXiv:0912.0847
Convergence of the calculationof successive transfer
Decomposition into successiveand simultaneous contributions
Various effective forces in the pairing channel have been proposed, with different features(finite/zero range, density dependence…)Unfortunately it is difficult to discriminate among them comparing with available data.
We want to follow a different strategy:
- Start from an Hartree-Fock calculation with a‘reasonable’ interaction. Then solve the pairing problemwith a bare interaction in the 1S0 channel. And finally addcorrelations beyond mean field. We know that these correlations strongly renormalizethe density of single-particle levels (effective mass) and their occupation factors (fragmentation), and we expectthat they can have a large effect on pairing properties.
Setting these findings in a broader context – pairing in heavy nuclei
Going beyond the quasi-particle approximation
by extending the Dyson equation…
to the case of superfluid nuclei (Nambu-Gor’kov), it is possible to consider both
and
J. Terasaki et al., Nucl.Phys. A697(2002)126
Renormalization of quasiparticles 120Sn
Argonne Argonne + induced interaction
h11/
2
Exp.
Th.
h11/2
h11/2
d5/2 3-
Semiclassical diagonal pairing matrix elements (120Sn)
Vind
Vbare
VGogny
F. Barranco et al., Phys. Rev. 72(2005)054314
E
E
E
E
(Vlow-k)
Renormalized pairing gaps
Local approximation
Argonne+ induced
Argonne
Induced
The pairing gap associated with the bare interaction is surface peaked;the induced interaction reinforces this feature
Microscopic justification of surface peaked, density-dependent pairing force
A. Pastore et al., Phys. Rev. C78 (2008) 024315
T. Duguet, T. Lesinski, A.Schwenk, in preparation
Mean field calculation with Vlow-k pairing force: 3-body force reduces the pairing gaps
According to a dynamical model of the halo nucleus 11Li, a key role is played by the coupling of the valence nucleonswith the vibrations of the system.The structure model has been tested with a detailed reactioncalculation, comparing with data obtained in a recent (t,p) experiment. Theoretical and experimental cross section are in reasonable agreement.
The exchange of collective vibrations represents a source of pairing also in heavy nuclei. The quantitative assessment of this contribution isan open problem.
T. Duguet et al., arXiv:0809.2895 and Catania Workshop
Vlow-k with SLy5 mean field
T. Roger, Ph.D. Thesis (2009)
Going beyond mean field: medium polarization effects
Self-energy
Induced interaction (screening)
Coupling of vibrations to single-particle motion
Effective mass mω
Increased level density at the Fermi energy
e1
e2
e1
=
1-MeV 3 N(0)
MeV2
5.1
mm22 MeV3.0V
2
21
2
)(
V
ee
V
Self-energy and effective mass
mVN
m
2)0(1
e2e2
e1
e1
=
Pairing from exchange of vibrations (induced interaction)
)(2 211
2
eee
V=
)( 21
2
ee
V
2V
indV =
22V - 0.3 MeV
Two time orderings
),()2(
1),( 1212
33
12 rRerdkR CMrki
CM
Pairing field in momentum space
Gogny Argonne
Local approximation
Argonne+ induced
Argonne
Induced
Effective masses
Bare (Argonne) gaps Renormalized gaps
Two-step effects : how important are they?
Excitation of ½- state following transfer
(t,p) transfer experiments: citations
2n transfer theory: 2nd order DWBA
A more phenomenological approach: particle-vibration matrix elements derivedfrom properties of experimental surface vibrations
Spectroscopic factors: overlap between 11Be and 12Be